The purpose of this talk is to prove an inequality
1 + x \le e^x, x \in R (1)
on an extension line of the construction of e. The distinction of our
method is to prove the inequality (1) without using the differential
calculus. As is shown, the inequality(1) characterizes the exponential
funtion $e^x$. Furthermore, we can prove some fundamental properties of $e^
x$ by using (1) and find that (1) is extremely important inequality with
respect to the exponential functin $e^x$.